Depending on your background, you probably don't have enough information to pick a long-term goal anyway.
I am afraid that sounds curmudgeonly, but I have also seen students shoot themselves in the foot because they decided they didn't need a class for their not very well informed goals.
>>Depending on your background, you probably don't have enough information to pick a long-term goal anyway.
Nah, it's totally possible for newbies to pick high-level long-term goals.
This can be something like "I want to teach my computer to tell apart dogs and cats", or "I want to create a website where people can buy and sell yarn." From there, Google searches can direct someone towards concepts and various methods of learning them.
I mean, you can disagree all you want, but this is in fact how many people learn things.
The two of you are talking about different things. What forkandwait is talking about is the propensity for people with only an undergraduate education in math (or less) to not actually know what a worthwhile goal is. They usually either lack the mathematical maturity to intuit how difficult a particular problem is (whether it's tractable with available mathematics, whether it's tractable for their ability, etc); or they formulate problems which are "not even wrong."
Of course this is in the context of choosing research problems to strive towards in math. If you tasked yourself with solving an open problem in math, it's more likely than not that, without any collaboration, you'd have no idea how to even work towards the goal due to all the unknown unknowns. If your goal is something concrete that can be augmented with mathematics, then yes I agree that goal setting can be useful. It doesn't take a volume of missing domain knowledge to develop that kind of goal.
Not in math. Unless you are a mathematician, I challenge you to pick a math equivalent of "I want to create a website where people can buy and sell yarn." I’ll wait.
People just assume learning math is the same as learning everything else. That is not even remotely true.
Genuinely curious - why do you think that? I have been self studying math for about 4 years now and find it to be the same as everything else that's worth learning: hard! But I haven't found that it's some entirely different realm divorced from all other intellectual pursuits.
I don’t really have time to give a thoughtful answer (it would be quite long), but the exact post you responded to gave an obvious difference. To roughly summarize that difference, producing anything of value in mathematics requires learning a tremendous amount of prior art, and without a tremendous amount of work you won’t even know what’s of value. It’s no wonder that many crackpots choose to work on high profile number theory problems, like Goldbach’s conjecture and previously Fermat’s Last Theorem, since the formulations are simple enough for laypeople to understand, yet the theories behind developed over hundreds of years are incredibly deep.
> everything else that’s worth learning: hard!
I disagree. I’ve learned many things worth learning that are not hard at all, but to each their own.
